A unified weighted inequality for fourth-order partial differential operators and applications

TL;DR

This paper establishes a fundamental weighted inequality for fourth-order PDE operators, enabling new Carleman estimates and resolvent bounds that lead to stabilization results for plate equations with various boundary conditions.

Contribution

It introduces a unified exponential-type weighted inequality for fourth-order operators, extending to singular weights, and applies it to derive Carleman estimates and stabilization results.

Findings

Established a fundamental inequality for fourth-order PDE operators.

Derived Carleman estimates under various boundary conditions.

Obtained resolvent estimates implying stabilization of plate equations.

Abstract

In this paper, we establish a fundamental inequality for fourth order partial differential operator $\cal P=\alpha\partial_s+\beta\partial_{ss}+\Delta^2$ ($\alpha, \beta\in\mathbb{R}$) with an abstract exponential-type weight function. Such kind of weight functions including not only the regular weight functions but also the singular weight functions. Using this inequality we are able to prove some Carleman estimates for the operator $\cal P$ with some suitable boundary conditions in the case of $\beta<0$ or $\alpha\neq 0, \beta=0$. As application, we obtain a resolvent estimate for $\cal P$, which can imply a log-type stabilization result for the plate equation with clamped boundary conditions or hinged boundary conditions.